Transforming a linear congruence equation into an equivalent one

Question

Is the equation $x \equiv -6 \bmod 5$ identical to $x \equiv 4 \bmod 5$ or to $\equiv 1 \bmod 5$?

Generally what is the best way to convert negative constant into positive? Do we have a formula for it?

Answer 1

You can add 5 without changing value. So mod 5, $-6 \equiv -1 \equiv 4$.

Answer 2

By definition of congruence $\pmod n$ $$ x \equiv y \pmod n \Longleftrightarrow \exists \ m \ s.t. \ x-y =m*n$$

Clearly all the letters denote integers. So it is a consequences of definition that adding multiple of $n$ doesn't change anything.

in fact you can view congruence mod a number like viewing and treating such number as $0$. Obviously is not formal but it gives some hints about what operations are legit in this case

Usually these are called the remainders classes, remainders given by division by $5$ in your case. Note that given an integer $x$, if $$x=q*5 + r$$ where $r$ is the remainder, then $$x+5*m=(q+m)*5+r$$ so $r$ Is again the remainder, showing that adding or subtracting $5$ (in this case) doesn't change anything in terms of the remainder

Answer 3

For an intuitive understanding,

$$x ≡ -6\mod5$$ means that $x=5k-6$ for some integer $k$. Now, we can substitute $k=n+2$ for some other integer $n$. Therefore, $x=5n+4$ and applying modulo,

$$x ≡ 4 \mod 5$$